Keep going.
But not here, in the forum.Factor completely ...
You already know that it is, so please explain why you asked.... (x^3 + 1) is [a] sum of cubes, right?
You already know that it is, so please explain why you asked.
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I have no clue what this question even means.Factor completely.
x^4 + x^3 + x + 1
What about factor by grouping?
x^3(x + 1) + (x + 1)
(x^3 + 1)(x + 1)
The binomial term (x^3 + 1) is the sum of cubes, right?
I have no clue what this question even means.
Any polynomial of degree 2k with real coefficients can be factored into k quadratics with real coefficients by the fundamental theorem of algebra, but that theorem does not tell you how to find such quadratics. It merely says that they exist. Moreover, such a factoring may well not be unique. Additionally, by that same fundamental theorem of algebra, such a polynomial can be factored uniquely (except for order) into 2k linear terms with complex coefficients. Consequently, the question as given makes no sense at all.
As you have recognized, there is a factor that is a sum of cubes with real coefficients, which can be factored into a linear term and a quadratic term, each with real coefficients. So the desired answer could be the product of two quadratics, each with real coefficients, or the product of one quadratic and two linear terms, all with real coefficients, or the product of four linear terms, perhaps some with complex coefficients. Which of those does "fully" mean?
Well, no; this thread is about a question that says to "factor completely", not to find the lowest common multiple. I'm not sure why you referred to LCM in post #5. I suspect that is a sort of cross-contamination from other threads. You meant "x^4 + x^3 + x + 1 = ...".
Factoring completely, in this context, presumably means to factor into factors of the lowest possible degree over the integers (or possibly over the reals or the complex numbers; the context is needed to be sure). You correctly did this: (x + 1)^2 (x^2 - x + 1).
Please stop changing the subject.... I asked about factoring completely ...
... (x^3 + 1) is the sum of cubes, right?
Please stop changing the subject.
You specifically asked whether x^3+1 is a sum of cubes.
You already knew that it is, so why did you ask?
I think you're fibbing.